If a measurement is made to determine th number of photons in mode "m"
a) What is going to be the expected result?
b) What is going to be the variance?

So, I was taking a look at it and I realized (or at least I think) that this might be a pure state because all the photons reside in a single mode. So, we could find the density operator, [tex]\rho[/tex], and it will be either one or zero since [tex]\rho=\rho^2 [/tex]. So, the mean will equal Tr([tex]\rho[/tex]n) however that will equal infinity which is does not seem to make sense. Then, alternatively I tried deriving another expression for the density operator- using some crazy assumptions such as [tex] (a^+)^2 |n>= \sqrt((n+1)(n+2))|n+2> [/tex] so that is what I have now and I don't know what to do with this...

So, what am I missing here? Is this really a pure state or is this a mixed state? Or do you have to assume something else? My main problem is dealing with the |n+2> part which is greatly bothering me. Once I get the mean it should be straightforward to find the variance. Any hints and comments would be greatly appreciated.

If you were given that the state was [tex] |\psi> = \sqrt(3)/2|2>_{m} + 1/2|4>_{m} [/tex]
would you be able to calculate the expectation value of N?
What if the state was [tex] |\psi> = \sqrt(3)/2|34>_{m} + 1/2|36>_{m} [/tex]?